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THE A-POLYNOMIAL HAS ONES IN THE CORNERS

Published online by Cambridge University Press:  01 March 1997

D. COOPER  and
D. D. LONG
D. COOPER
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106, USA
D. D. LONG
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106, USA
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Abstract

1. Definition of the A-polynomial

The A-polynomial was introduced in [3] (see also [5]), and we present an alternative definition here. Let M be a compact 3-manifold with boundary a torus T. Pick a basis λ,μ of π1T, which we shall refer to as the longitude and meridian. Consider the subset RU of the affine algebraic variety R = Hom (π1M, SL2[Copf ]) having the property that ρ(λ) and ρ(μ) are upper triangular. This is an algebraic subset of R, since one just adds equations stating that the bottom-left entries in certain matrices are zero. There is a well-defined eigenvalue map

formula here

given by taking the top-left entries of ρ(λ) and ρ(μ).

Type
Research Article
Information
Bulletin of the London Mathematical Society , Volume 29 , Issue 2 , March 1997 , pp. 231 - 238
Copyright
© The London Mathematical Society 1997

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